Planet layouts

You do need approval buildings in the early stages of the game,but they are easily demolished later on when you get starbases with approval upgrades up and running.

You might be right about farms, although i usually have at least 3 farms even on manufacturing and science worlds.... because its all about population storage (for invasions) up to the maximum comfortable levels that approval allows.

For wealth/research, I've been going with a 3-2-1 strategy: 3 research wealth, 2 farms, 1 factory (to assist building time). Maintain that ratio. There are too many factors to say whether this is optimal, but in my empirical tests it worked out near the top in terms of output/time.

Being an old g2 player, i am very used to building only markets, because its a financial black hole that game,,,, pretty much all that is allowed is only 1 or two science and production worlds or you go broke.

Building one factory on every world is a totally alien concept too me! lol

Nah. More Farms = Better. People are really underestimating farms in here. 

Population directly increases all three of your production types, even without their respective improvements present. Want to put all of your production into manufacturing without taking a huge hit to maintenance costs? Increase your population. High population will pretty much negate all maintenance costs even with little to no production sent to wealth. Want to have a research or manufacturing planet generate money while you are not building or research? Increase your population. Even without wealth improvements, a high population will still generate you hundreds of bc per turn. Same thing for research. Your planets are going to have subpar production values but high maintenance with only one farm because your population cap would be too low. Furthermore, a high population makes makes it harder for enemies to take over your planet in one try.

Consider if you ever have no need to build or research anything, all your factories and research are going to be going to waste (unless you waste time to demolish them and rebuild). With a high pop, you can easily redistribute your production and still achieve meaningful numbers, regardless of the improvements you have.

Approval improvements are things you generally only need mid game, with the exception of Ideology generating improvements. Early game, your pop is going to be too low to be bothered, and late game, you just need some relics, star bases, and a planet with that Virtual World improvement. Approval is not really a major thing to worry about, however. As long as your smily face is not in the red, you are good to go, and when it is nearing the read, research some Approval/Morale, and that will help you along early on until you get better established.

Your best approach should be to balance your tiles between a specialization and farms, leaving a few tiles left over for other things, such as Approval or Ideology when you need it. Go nuts on class 20+ planets. On research planets, a good layout should also take into consideration the day when you are done researching everything. Additionally, wealth should be much lower on your priority list. You can easily operate at a lost for a long time by selling your tech and treaties to friendly factions or minor races. The real money sinkhole is in ships, so avoid building them unless you really have to (e.g. war with neighbor).

I generally like to go with 7 tiles of farms (to make one large hex) and 7 tiles of a specialization where possible. Leftover tiles will be distributed as the situation calls, but usually for Approval or Ideology. On class 25+ planets, I do 9 and 9 instead.

There is some misleading information in the above post. But regardless the "mostly farms" strategy is viable and better than what the AI can do, so more power to you. Focusing provides superior numbers, if one can work out how to manage them well.

I have 4-5 tiles per planet on farms.  I have 1-2 devoted to manufacturing.  Approval is in flux with the loss of LEP, but it looks like 2, one entertainment and one missionary. The rest is specialization as solid and adjacent as possible.  One of the specialties is manufacturing.  The overall principle is to keep population growing and pushing towards an increasing food cap and pump all that production into multipliers.  There is also a subjective feel for the ratio of specialty types you need in a given game.  For me, it is all about having enough wealth planets to pay for all the others.  I am never convinced I have it right until suddenly I am swimming in money.  There has to be a better way to do that.  If it is a lesser quality planet, there may be only one multiplier building.  Less than that, it is a default low level manufacturing planet aiding another planet's shipyard. So far, this is dominating the AI economically on genius difficulty, although sometimes with early struggles due to laziness about building military.  It also seems to fit well with a focus-based economy.  I win too early with technology victories to get there, but I can see tearing out the manufacturing buildings on research and wealth planets in late game as a good strategy to apply to this method.  

I offer all this as one example, not sage advice.  I am learning. Also, I am abusing the heck out of a custom faction with my favorite traits and a Thalan tech tree.  Their hives just fit my play style all too well.  I am sure that plays a factor in my decisions, but I couldn't separate it out from all the other factors.  

The general equation for a fully-populated planet's overall output is

O = (1 + c) * (1 + p) * (P + (1 + f)*(Bf + Nf * F)) * (m * (1 + Bm + bm * Nm) + r * (1 + Br + br * Nr) + w * (1 + Bw + bw * Nw))

where O is the total output of the planet, c is the coercion modifier, p is the planet's percentile production bonus, P is the planet's flat production bonus, f is the planet's percentile food bonus, Bf is the planet's flat food bonus from all sources other than basic food improvements ('farms;' e.g. Xeno Farm), Nf is the number of farms, F is the average amount of food provided per farm, m is the fraction of planetary production allocated to manufacturing, Bm is the planet's percentile bonus to manufacturing output from all sources other than basic manufacturing improvements ('factories'), bm is the average bonus to manufacturing output per factory, Nm is the number of factories on the planet, r is the fraction of the planet's production allocated to research, w is the fraction of the planet's production allocated to wealth, and Br, br, Nr, Bw, bw, and Nw are the terms for research and wealth corresponding to the similarly-named manufacturing-related terms.

Under the assumption that you're using a fully-specialized planet (i.e. all of your planet's production goes towards a single output type), it's actually fairly easy to compute the long-term optimal build of a planet (note: 'easy' is a relative term; this is not a difficult equation to solve, but using a rule of thumb such as 1 farm per factory is even easier, though often less optimal - that is, after all, the trade-off for using a rule of thumb instead of computing what's best for the given scenario, as a rule of thumb gives you a 'good enough' solution but not necessarily the 'best' solution). In such a scenario, the equation governing the planet's output boils down to

O = (1 + c) * (1 + p) * (P + (1 + f) * (Bf + N * F)) * (1 + Bm + B * (T - (1 + A) * N))

where c is the coercion penalty, p is the planet's total production bonus, P is the planet's flat production bonus, f is the planet's total food bonus, Bf is base amount of food that the planet, N is the number of farms you build, F is the average food granted by each farm, Bm is the planet's bonus to the output type you chose from all sources other than basic factories, T is the total number of tiles available on the planet for basic factories of the type corresponding to the desired output and morale improvements and farms, A is the average number of approval structures you build per farm, and B is the average bonus you gain per basic factory of the type corresponding to the desired output. (Note that in the section covering the fully-specialized planet, 'factories' are basic output improvements, not specifically basic manufacturing improvements.) This is a concave-down parabola in N, so if you make a few simple assumptions about the average bonus per farm and per factory, you should have everything you need to solve for the value of N which maximizes the equation. That value of N can be found as

N = [(1 + Bm + B * T) * (1 + f) * F - (P + (1 + f) * Bf) * (1 + A) * B] / [2 * (1 + f)*(1 + A) * F * B]

For example, if T = 10, F = 2, B = 0.15, Bf = 5, P = 5, and all the other constants are 0, then the optimal number of farms N to build on the planet is 5.8. Obviously you cannot build exactly 5.8 farms, but either 5 or 6 farms should get you pretty close to the theoretical maximum. Note that this is long-run optimal, i.e. once the planet hits its population cap this is the configuration which maximizes the planet's potential output per turn. Ideally speaking, a fully-specialized planet will eventually neither have nor need improvements giving a bonus to an off-type output, as the population will be sufficiently high that the unmodified production allows improvements to be built or upgraded at a reasonable pace and of course once the planet is fully built the off-type improvements are no longer useful (at least until the next upgrade cycle comes around). Note that all constants should be expressed as decimals or fractions (i.e. a 10% bonus to something is 0.1 or 1/10, not 10) and should be positive if the modifier is a bonus and negative if the modifier is a penalty.

If you wanted to do so, you could use one of the above equations (or a more generalized output equation which accounts for all three output types, though maximizing the three-output equation is much more difficult) to solve for the long-run optimal planet builds. You could also use these to work out the sums required to determine the build which is optimal over the next N turns.

Other configurations can give you more output in the short term. For example, a farm is useless until the planet's population reaches a point where a farm is required to allow the population to continue to grow at the current rate, and additionally the farm will not provide its full bonus until several turns after that point as it will take time for the population to grow to the new limit; thus, there is often some number of turns over which the total output generated by the planet would have been greater had you built a factory rather than a farm, even though it may still be more optimal in the long run to build a farm.

I would suggest that you should use a rule of thumb to get a 'good enough' solution rather than working out exactly what the optimal solution for your particular planet happens to be unless you happen to like running a computation for every planet (and also every time the improvements you have access to change), but the above should give you all that you need to know if you desire to determine the optimal builds for a planet.